User feldmann denis - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:29:28Z http://mathoverflow.net/feeds/user/17164 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119696/translation-of-a-non-standard-analysis-formulation Translation of a non-standard analysis formulation Feldmann Denis 2013-01-23T21:39:41Z 2013-01-30T11:42:37Z <p>Usually, it is quite easy (if cumbersome) to translate a formula like "if $\varepsilon$ is infinitesimal, and if $f$ is differentiable at $a$, $f'(a)$ is the shadow of $\frac{f(a+\varepsilon)-f(a)}{\varepsilon}$". But I cannot find a standard translation of the conjecture :"There exists a galaxy of non-standard integers containing an infinite number of primes", where galaxies are the equivalence classes of $\mathbb N$ by the relation $m\simeq n \iff (m-n)$ is standard. The closest I can find is the k-tuple conjecture (the first of the Hardy-Littlewood conjectures), but it seems obvious this is quite stronger than mine.</p> http://mathoverflow.net/questions/112199/differential-equations-and-axiom-of-choice Differential equations and axiom of choice Feldmann Denis 2012-11-12T17:34:43Z 2013-01-23T14:36:37Z <p>In the most general context, the Picard-Lindelöf theorem (aka Cauchy-Lipschitz in French) asserts the existence of a maximal solution for $\dot{x}(t) = f(t,x(t))$, i.e. of a solution $x(t)$ defined on a interval $I$ such that there exist no other solution whose restriction to $I$ coincide with $x$. The usual proofs of this (when $f$ is such that there is no local unicity) use Zorn's lemma, or some other weaker form of choice. But is this result actually not provable in ZF?</p> http://mathoverflow.net/questions/94998/integration-in-the-surreal-numbers Integration in the surreal numbers Feldmann Denis 2012-04-24T06:04:47Z 2012-12-27T19:00:07Z <p>In the appendix to ONAG (2nd edition), Conway points that the definition of integration (using Riemann sums as left and right options) gives the "wrong" answer : $\int_0^\omega \exp(t)\thinspace dt=\exp(\omega)$ (instead of $\exp(\omega)-1$). I wonder if this was not due to lack of some options (presumably right ones), and if a better definition could not be sought by using Kurzweil-Henstock integration (but I was not able to concoct one, of course, else I would not ask here). Has the idea already been tried?</p> http://mathoverflow.net/questions/116848/aronszajn-trees-and-the-transfinite-subway Aronszajn trees and the transfinite subway Feldmann Denis 2012-12-20T10:15:40Z 2012-12-20T13:07:47Z <p>The transfinite subway puzzle (see <a href="http://mathforum.org/kb/message.jspa?messageID=229112" rel="nofollow">http://mathforum.org/kb/message.jspa?messageID=229112</a>) is one of those clever puzzle only mathematicians can enjoy (other ones being the blue-eyed islanders puzzle (see <a href="http://terrytao.wordpress.com/2011/04/07/the-blue-eyed-islanders-puzzle-repost/" rel="nofollow">http://terrytao.wordpress.com/2011/04/07/the-blue-eyed-islanders-puzzle-repost/</a> ), or the use of axiom of choice to guess almost always the content of all but a finite number of boxes, see (in French) <a href="http://www.madore.org/~david/weblog/2009-05.html#d.2009-05-16.1643" rel="nofollow">http://www.madore.org/~david/weblog/2009-05.html#d.2009-05-16.1643</a>). My question (about the tranfinite subway) is : is there any relationship between this strange property of $\omega_1$ (i.e. that the subway arrives always empty) and the existence of $\aleph_1$-Aronszajn trees ?</p> http://mathoverflow.net/questions/116445/positive-results-coming-from-paradoxes Positive results coming from paradoxes Feldmann Denis 2012-12-15T12:11:12Z 2012-12-16T06:24:25Z <p>Many examples comes to mind, the most famous being the Gödel's theorems viewed as formalisations of the Liar's paradox. I just realised that the proof of non-calculability of Kolmogorov complexity is a positive rewriting of Berry's paradox. My question (perhaps to be made into collective mode) is a) what are the best examples you know ? b) (more important) is there some explication of this productivity of paradoxes (or, conversely, do you know of paradoxes with no interesting follow-up) ?</p> http://mathoverflow.net/questions/114383/examples-where-adding-complexity-made-a-problem-simpler/114479#114479 Answer by Feldmann Denis for Examples where adding complexity made a problem simpler Feldmann Denis 2012-11-26T02:43:27Z 2012-11-26T02:43:27Z <p>Quasi-periodic tilings as projections of periodic ones from a higher-dimensional space</p> http://mathoverflow.net/questions/114184/a-question-about-the-axiom-of-choice-and-straight-lines-in-the-euclidean-plane/114187#114187 Answer by Feldmann Denis for A question about the Axiom of Choice and straight lines in the Euclidean plane. Feldmann Denis 2012-11-22T20:46:42Z 2012-11-22T22:00:58Z <p>Let $C$ the curve defined by $y=x^x$ (for $x>0$) union the point $x=0,y=1$. The set of tangents to $C$ (including the $y$ axis) satisfies the hypothesis, as the slope of those tangents grow continuously from $-\infty$ to $+\infty$, and it is impossible to draw three distinct tangents from any point of the plane. So what is wrong with this example? </p> <p>I just forgot the union of the lines must be $E$ . Time to go to bed ; please delete this answer if you can. On the other hand, in this case, the union of the lines is a closed subset of $E$, so it may still be worth something</p> http://mathoverflow.net/questions/66075/the-half-life-of-a-theorem-or-arnolds-principle-at-work/112446#112446 Answer by Feldmann Denis for The half-life of a theorem, or Arnold's principle at work Feldmann Denis 2012-11-15T03:21:56Z 2012-11-15T03:21:56Z <p>The discovery of the Mandelbrot set by Udo of Aachen circa 1250 should perhaps count for something, no ? (see <a href="http://en.wikipedia.org/wiki/Udo_of_Aachen" rel="nofollow">http://en.wikipedia.org/wiki/Udo_of_Aachen</a>).</p> http://mathoverflow.net/questions/112026/characteristic-polynomial-of-hypercube-graph Characteristic polynomial of hypercube graph Feldmann Denis 2012-11-10T21:11:40Z 2012-11-12T04:11:26Z <p>This is probably well-known, but... Define the $n$-dimensional hypercube graph $H_n$ as having for vertices the integers between 0 and $2^n-1$, and edges between integers differing by a power of 2. The characteristic polynomial of $H_n$ is then $\prod_{k=0}^n(x-n+2k)^{\frac {n!}{k!(n-k)!}}$, i.e. $(x-3)(x-1)^3(x+1)^3(x+3)$ for a cube, $(x-4)(x-2)^4x^6(x+2)^4(x+4)$ for a tesseract, etc. Is there a graph-theoretic proof of this result? </p> http://mathoverflow.net/questions/111746/is-there-a-closed-form-for-the-characteristic-polynomial-of-the-graph-cycle-of Is there a closed form for the characteristic polynomial of the graph cycle (of n edges and n summits) ? Feldmann Denis 2012-11-07T18:38:08Z 2012-11-08T01:25:25Z <p>Is there a closed form for the characteristic polynomial of the graph cycle (of $n$ edges and $n$ summits)? </p> <p>I know it for the graph path (it is a Chebyschev polynomial), but I couldn't find a closed form when adding the missing edge.</p> http://mathoverflow.net/questions/109976/tarski-monster-groups-which-ones-dont-exist Tarski monster groups : which ones dont exist ? Feldmann Denis 2012-10-18T05:20:43Z 2012-10-23T12:07:53Z <p>Is there any large $p$ for which it is proven that an infinite group with all non-trivial subgroups cyclical of order $p$ doesn't exist? And (if there is), which is the largest such $p$ ? All I could find is the $10^{75}$ upper bound, but I admit I didn"t look very far...</p> http://mathoverflow.net/questions/109990/are-all-compact-groups-amenable Are all compact groups amenable ? Feldmann Denis 2012-10-18T09:16:38Z 2012-10-18T09:37:21Z <p>Wikipedia states that the Haar measure on a compact group is a mean (and that every compact group is amenable). But, obviously, the Haar mesure on the group of unit quaternions cannot be defined on every subset, else the Banach-Tarski paradox would not happen. What am I missing?</p> http://mathoverflow.net/questions/108707/banach-tarski-vs-von-neumann Banach-Tarski vs von Neumann Feldmann Denis 2012-10-03T14:09:02Z 2012-10-03T14:09:02Z <p>While not so well known, the <a href="http://en.wikipedia.org/wiki/Von_Neumann_paradox" rel="nofollow">von Neumann paradox</a> is built among the same lines, in dimension 2 and with transforms within the special linear group. But what is wrong with the following "proof" : it is not very hard to show that the group generated by the two rotations of angle $\alpha$ around $O$ and $C$, with $OC&lt;1/100$, say, and $\alpha/\pi$ irrational and small (say $&lt;1/100$ too), is non-abelian and free. The von Neumann construction applied to two disjoint unit discs then split them in four sets (plus a few fixed points) and sends those sets injectively, by rotations and translations, to disjoint sets included in the union of four copies of the unit disc, this union having something like 1.04 area of the disc. Obviously, this is very wrong, as the Banach-Tarski construction of a full additive measure invariant by isometries of the plane preclude such a thing. Where am I mistaken?</p> http://mathoverflow.net/questions/93828/how-large-is-tree3 How large is TREE(3) ? Feldmann Denis 2012-04-12T06:35:54Z 2012-09-27T11:46:55Z <p>Friedman, in <a href="http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdf" rel="nofollow">http://www.math.osu.edu/~friedman.8/pdf/EnormousInt112201.pdf</a>, shows that TREE(3) is much larger than n(4), itself bounded below by $A^{A(187195)}(3)$ (where $A$ is the Ackerman function and exponentiation denotes iteration). But actually, using the fast-growing hierarchy, $n(p)$ is smaller than $f_{\omega^{\omega^\omega}}(p)$ (shown by Friedman in <a href="http://www.math.osu.edu/~friedman.8/pdf/finiteseq10_8_98.pdf" rel="nofollow">http://www.math.osu.edu/~friedman.8/pdf/finiteseq10_8_98.pdf</a>), while it seems that TREE grows faster than $f_{\Gamma_0}$ (${\Gamma_0}$ being the Feferman-Schütte ordinal). So it could well be that in fact TREE(3) is larger than, say, n(n(4)), or even any number expressible by iterations of n. What is known on this question?</p> http://mathoverflow.net/questions/107155/what-grows-faster/107160#107160 Answer by Feldmann Denis for What grows faster? Feldmann Denis 2012-09-14T09:39:04Z 2012-09-14T10:00:54Z <p>Much too easy for this site, and already answered. Anyway, BusyBeaver grows faster that any computable function (almost by definition); as the other two are computable... (btw, TREE grows inimaginably faster that the recursive functions you cite next). All these questions tend to show that you have not studied the fast-growing hierarchy : in this notation, Conway's $n\rightarrow n \rightarrow\dots\rightarrow n$ (with $n$ arrows) is (much) smaller than $f_{\omega^2+1}(n)$, and any "recursive" construction in the line of BEAF grows slower than $f_{\omega^\omega}$, while TREE grows much faster than $f_{\epsilon_0}$, or even $f_{\Gamma_0}$... </p> http://mathoverflow.net/questions/107043/is-monskys-theorem-depending-on-axiom-of-choice Is Monsky's theorem depending on axiom of choice ? Feldmann Denis 2012-09-12T20:55:41Z 2012-09-12T21:21:28Z <p>The extension of the 2-adic valuation to the reals used in the usual proof uses clearly AC. But is this really necessary ? After all, given a equidissection in $n$ triangles, it is finite, so it should be possible to construct a valuation for only the algebraic numbers, and the coordinates of the summits (with a finite number of "choices"), and then follow the proof to show that $n$ must be even. Or am I badly mistaken ? </p> http://mathoverflow.net/questions/106448/topologies-on-the-field-of-rationals Topologies on the field of rationals Feldmann Denis 2012-09-05T16:44:35Z 2012-09-06T14:06:03Z <p>Ostrowski's theorem give the answer for valuations, but is there a complete classification of (at least separated) topologies on Q (compatible with the field operations, obviously)?</p> http://mathoverflow.net/questions/105758/homeomorphism-of-the-rationals/105763#105763 Answer by Feldmann Denis for Homeomorphism of the rationals Feldmann Denis 2012-08-28T21:23:01Z 2012-08-28T21:23:01Z <p>Not a really complete answer, but look at the question mark fonction (see <a href="http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function" rel="nofollow">http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function</a>) to a counter-example satisfying 1,2,3 and 4 : $?(\sqrt2)=7/5$, for instance.</p> http://mathoverflow.net/questions/105715/functions-with-null-derivative Functions with null derivative Feldmann Denis 2012-08-28T13:26:37Z 2012-08-28T13:56:45Z <p>I am not here referring to <a href="http://en.wikipedia.org/wiki/Cantor_function" rel="nofollow">the devil staircase</a>, but to <a href="http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function" rel="nofollow">the question mark function</a>. This is a strictly increasing function from $\mathbb{Q}$ to $\mathbb{Q}$, with derivative always $0$. I have two questions: </p> <ol> <li><p>Is this a surprising result (in the same way that continuous functions nowhere differentiable were thought surprising when discovered)? </p></li> <li><p>What are the properties preventing this (obviously, we need a topological field to speak of derivative; are there exemples of connected topological fields with non constant functions having everywhere a null derivative?)</p></li> </ol> http://mathoverflow.net/questions/105555/integrating-1-xsin1-x/105563#105563 Answer by Feldmann Denis for integrating 1/(x*sin(1/x)) Feldmann Denis 2012-08-26T18:09:41Z 2012-08-26T18:09:41Z <p>Try Liouville theorem ; as the obvious $u=1/x$ lend to integration of $\frac 1{u\sin u}$, it seems impossible indeed in elementary terms ; actually, Maple isn't even able to integrate it using special functions</p> http://mathoverflow.net/questions/105222/hales-work-on-kepler-conjecture Hales work on Kepler conjecture Feldmann Denis 2012-08-22T10:28:24Z 2012-08-22T14:33:31Z <p>It seems that the Fulkerson prize has been attributed to Thomas Hales for this work. What is the present status of the conjecture, then?</p> http://mathoverflow.net/questions/103761/fast-growing-hierarchy-and-turing-machines Fast-growing hierarchy and Turing machines Feldmann Denis 2012-08-02T05:50:58Z 2012-08-13T07:10:41Z <p>Is it possible to get an estimate of the size of a Turing machine computing $f_\alpha(n)$, for a given $\alpha$ (I am especialy interested in moderately large $\alpha$ like the ordinal of Fefferman-Schütte, or the small Veblen ordinal)? The idea is to get an idea of the size of the BusyBeaver function $BB(n)$ for moderate values of $n$, as the litterature usually only mention the exact known values (for $n\le 6$) and the fact that such values will probably never be known for $n=10$, say.</p> http://mathoverflow.net/questions/95677/mandelbrot-set-and-analytic-functions-such-that-fazfz2c Mandelbrot set and analytic functions such that $f(az)=f(z)^2+c$ Feldmann Denis 2012-05-01T16:29:57Z 2012-08-03T20:06:50Z <p>It is well known that the function $f(z)=2\cos(\sqrt {-z})$ (or more accurately the entire function $f(z)=2\sum_{n=0}^\infty \frac{z^n}{(2n)!}$) satisfies such a functional equation, i.e. $f(4z)= f(z)^2-2$ ; it is not hard to show that this is the unique solution with $c=-2$ and $f'(0)=-1$. Patient calculations (see for instance <a href="http://denisfeldmann.fr/PDF/mandel.pdf" rel="nofollow"> this note in french</a>), or the classical results of Tan Lei, show that this implies that the small cardioids on the left main antenna of the Mandelbrot set (on the real axis near $-2$) are in position $-2+3\pi^2/2^{2n+1}+o(4^{-n})$. It would be easy to generalise this result to other Misiurewicz points, if similar functions were known for other values of $c$, as the small copies of the $M$-set lie similarly near the zeros of these functions (after appropriate rescaling) ; it is not hard to show their existence and unicity, but have they been studied, and what is known on their zeros?</p> http://mathoverflow.net/questions/101996/cantor-theorem-on-orders Cantor theorem on orders Feldmann Denis 2012-07-11T21:53:24Z 2012-07-18T06:44:36Z <p>It is "a well-known theorem of Cantor", said Sierpinski (circa 1920), that every countable total order can be imbedded in the rationals, and he proceeds to demonstrate that, assuming the continuum hypothesis, it is possible to construct a similar "universal order" of cardinal $\aleph_1$. I have two questions : 1) Where did Cantor prove his theorem 2) Can CH be weakened in the previous result ? For reference, Sierpinski construct an ordering on binary sequences indexed by countable ordinals (up to some ordinal $&lt;\omega_1$), having the desired property than any countable Dedekind cut is separated by some element ; this is actually the same order than the one on surreal numbers born before day $\omega_1$ (as easily shown by the Gondor construction) </p> http://mathoverflow.net/questions/101996/cantor-theorem-on-orders/101997#101997 Answer by Feldmann Denis for Cantor theorem on orders Feldmann Denis 2012-07-11T22:28:18Z 2012-07-11T22:51:43Z <p>Sorry, almost all pertinent answers (even with the exact reference of the Sierpinski article) were given in the article <a href="http://mathoverflow.net/questions/57597/universal-order-type" rel="nofollow">http://mathoverflow.net/questions/57597/universal-order-type</a> ; so the only question I have left is "where did Cantor state it ?" </p> http://mathoverflow.net/questions/99421/computational-complexity-of-calculating-the-nth-root-of-a-real-number/99430#99430 Answer by Feldmann Denis for Computational complexity of calculating the nth root of a real number Feldmann Denis 2012-06-13T11:05:23Z 2012-06-13T11:05:23Z <p>The Newton-Raphson algorithm uses, for computation of $A^{1/p}$, the sequence $u_0=A$, $u_{n+1}=u_n-\frac {u_n^p-A}{pu_n^{p-1}}$, whose speed of convergence , always quadratic, is essentially independent of $p$ (and $A$). So, mostly, it asks for $\ln p$ multiplications and 1 division at each step.</p> http://mathoverflow.net/questions/99171/is-the-sum-sinn-bounded/99172#99172 Answer by Feldmann Denis for Is the sum sin(n) bounded? Feldmann Denis 2012-06-09T11:24:34Z 2012-06-09T11:24:34Z <p>In this particular case, use $s_n(\theta)=\sum_{k=0}^n\sin(k\theta)=\Im(\sum_{k=0}^n\exp(ki\theta))=\dots=\frac{\sin(n\theta/2)\sin((n+1)\theta/2)}{\sin (\theta/2)}$</p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions Faa di Bruno's formula for inverse functions ? Feldmann Denis 2012-05-31T15:59:19Z 2012-06-02T17:17:24Z <p>It is easy to get a expression for the nth-derivative of an inverse fuction ; starting from $(f^{-1})'=\frac{1}{f'\circ f^{-1}}$, one gets things like $(f^{-1})^{(n)}=\frac{\sum a_k\prod (f^{(n_j)}\circ f^{-1})^j}{(f'\circ f^{-1})^{2n-1}}$, with reasonably easy constraints on the $n_j$. But what are the values of the $a_k$? I believe I read somewhere this was an application of umbral calculus, but I dont see how, and inverting Faa di Bruno's formula on the identity $f\circ f^{-1}=id$ dont seem to get anywhere.</p> http://mathoverflow.net/questions/98501/faa-di-brunos-formula-for-inverse-functions/98664#98664 Answer by Feldmann Denis for Faa di Bruno's formula for inverse functions ? Feldmann Denis 2012-06-02T15:47:49Z 2012-06-02T15:47:49Z <p>To precise my question, I was asking for the exact values of the $a_k$. Thanks to Tom Copeland, I could find the sequence A176740 of OEIS, giving a complete answer (with useful links) to this problem.</p> http://mathoverflow.net/questions/20765/the-problem-of-finding-the-first-digit-in-grahams-number/96295#96295 Answer by Feldmann Denis for The problem of finding the first digit in Graham's number Feldmann Denis 2012-05-08T05:21:43Z 2012-05-08T05:21:43Z <p>Similarly, it is 1 if the base is of the shape $3^n$, with $n$ a power of 3 (less than $\ln G$) ; other cases are easy for small powers of 3. On the other hand, Graham's number is much too large for the question to be really interesting ; the determination of the first digit in base 10 of $A(9,9)$ (where $A$ is the Ackermann function) should already be inaccessible. </p> http://mathoverflow.net/questions/126789/does-the-prime-number-theorem-prove-that-the-primes-cannot-be-exactly-identified Comment by Feldmann Denis Feldmann Denis 2013-04-07T16:56:44Z 2013-04-07T16:56:44Z Have you stop beating your wife ? I would appreciate at least a yes or no http://mathoverflow.net/questions/119913/what-is-the-difference-between-a-function-and-a-morphism Comment by Feldmann Denis Feldmann Denis 2013-01-26T05:59:15Z 2013-01-26T05:59:15Z Don't you really mean &quot;whose domain is $A$ and range $B$&quot; ? (then, of course, you have to think of the objects of your category as sets). But of course, the answer is negative, as different morphisms can correspond to the same function. http://mathoverflow.net/questions/119493/toroidality-testing Comment by Feldmann Denis Feldmann Denis 2013-01-21T21:14:17Z 2013-01-21T21:14:17Z Well, you can check if they dont have as minor one of the 16000 already known obstructions (for not too large graphs, this is less absurd than it sounds...) http://mathoverflow.net/questions/119494/examples-of-exotic-induction Comment by Feldmann Denis Feldmann Denis 2013-01-21T18:11:37Z 2013-01-21T18:11:37Z &quot;Proofs from the Book&quot; gives the nice ($P(1)$ and ($P(n)$ implies $P(2n)$) and ($P(n)$ implies $P(n−1)$)) implies $\forall n, P(n)$ as a way to prove the general inequality between geometric and arithmetic mean (due to Artin, or perhaps Cauchy) – Feldmann Denis 0 secs ago http://mathoverflow.net/questions/119267/associativity-of-infinite-series Comment by Feldmann Denis Feldmann Denis 2013-01-18T13:00:46Z 2013-01-18T13:00:46Z Inappropriate for MO. Think of cases like $a_n=1$ and $b_n=-1$. http://mathoverflow.net/questions/118242/have-discovered-a-recursive-formula-for-prime-density-is-this-known Comment by Feldmann Denis Feldmann Denis 2013-01-07T04:35:32Z 2013-01-07T04:35:32Z Something is very wrong here.If any formula of this shape was correct, it would give a fast way to calculate $\pi(x)$. Check it for not too small values of $n$... http://mathoverflow.net/questions/115787/how-much-is-10-powered-by-a-finite-number Comment by Feldmann Denis Feldmann Denis 2012-12-08T10:11:31Z 2012-12-08T10:11:31Z You should read the previous comments to your posts (the key word is &quot;read&quot;) http://mathoverflow.net/questions/115099/are-grothendieck-universes-enough-for-the-foundations-of-category-theory Comment by Feldmann Denis Feldmann Denis 2012-12-01T20:50:35Z 2012-12-01T20:50:35Z Actually, the Grothendieck axiom of universe U is much stronger than existence of a strongly inaccessible cardinal, being equivalent to the exoistence of a proper class of inaccessibles (but it is weaker than the existence of a 1-inaccessible cardinal) http://mathoverflow.net/questions/114959/counterpart-of-weierstrass-theorem/114986#114986 Comment by Feldmann Denis Feldmann Denis 2012-11-30T14:47:06Z 2012-11-30T14:47:06Z And I cannot resist to add a reference to the &lt;a href=&quot;<a href="http://en.wikipedia.org/wiki/Long_line&quot" rel="nofollow">en.wikipedia.org/wiki/Long_line&quot</a>; rel=&quot;nofollow&quot;&gt;long line&lt;/a&gt;, locally isomorphic to R, sequentially compact, but not compact. http://mathoverflow.net/questions/114234/relating-a-polynomial-equation-to-the-characteristic-equation-of-a-hermitian-matr/114236#114236 Comment by Feldmann Denis Feldmann Denis 2012-11-23T13:31:33Z 2012-11-23T13:31:33Z $\frac{F+F^H}2$ is indeed hermitian, but why should it have the same characteristic polynomial than $F$? http://mathoverflow.net/questions/114018/fastest-way-to-factor-integers-260/114028#114028 Comment by Feldmann Denis Feldmann Denis 2012-11-21T12:36:28Z 2012-11-21T12:36:28Z As Pari/gp has been developped by Henri Cohen, I would expect it is reasonably optimal for this kind of task... http://mathoverflow.net/questions/112026/characteristic-polynomial-of-hypercube-graph/112035#112035 Comment by Feldmann Denis Feldmann Denis 2012-11-10T22:34:56Z 2012-11-10T22:34:56Z Well, actually, I am looking for things like calculations of the characteristic polynomial by manipulations of the graph (addig edges or vertices, product of graphes, ans so on), and trying to get some intuition by using well-known families of graphs. But I may be completely on the wrong track http://mathoverflow.net/questions/111746/is-there-a-closed-form-for-the-characteristic-polynomial-of-the-graph-cycle-of Comment by Feldmann Denis Feldmann Denis 2012-11-08T16:33:16Z 2012-11-08T16:33:16Z Sorry for the misunderstanding. Yes, I wanted a closed form for the coefficients of the characteristic polynomial, but my main question was : is there a simple relation between the polynomials for graphs differing only by one edge, as in this case. http://mathoverflow.net/questions/110628/are-the-foundations-of-mathematical-logic-shaky Comment by Feldmann Denis Feldmann Denis 2012-10-25T08:29:14Z 2012-10-25T08:29:14Z And the preface to the logic part of Nicolas Bourbaki's treaty (Ens, ch.I) says &quot;we will not discuss here the possibility of teaching (or describing) mathematics to someone who couldn't read, write or count&quot; http://mathoverflow.net/questions/110345/does-a-power-series-converging-everywhere-on-its-circle-of-convergence-define-a-c/110453#110453 Comment by Feldmann Denis Feldmann Denis 2012-10-23T22:47:28Z 2012-10-23T22:47:28Z Probably, Gulbenkian thinks that the only reason a series has radius of convergence $r$ is that there must be a pole at this distance. Alas, other kinds of singularity (such as branch points) exists...